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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 459.10002

**Autor: ** Erdös, Paul

**Title: ** Problems and results in number theory. (In English)

**Source: ** Recent progress in analytic number theory, Symp. Durham 1979, Vol. 1, 1- 13 (1981).

**Review: ** [For the entire collection see Zbl 451.00001.]

In the author's words, ``somewhat unconventional problems on sieves, primes and congruences'' are discussed. Of the wealth of the problems, let us take a few examples. The integer n is said to be a barriere for an arithmetic function f if m+f(m) \leq n for all m < n. Question: are there infinitely many barriers for \epsilon\upsilon(n), for some \epsilon > 0? Here \upsilon(n) denotes the number of distinct prime factors of n. A related problem: is it true that **lim**_{n ––> oo}**max**_{m < n}(m+d(m))-n = oo? There are also many problems concerning consecutive primes. Let d_{n} = p_{n+1}-p_{n}. To prove that for any c > 0 there are infinitely many k such that d_{k} > c log k log log k log log log log k/(log log log k)^{2} (10000 dollars offered for a proof).

**Reviewer: ** M.Jutila

**Classif.: ** * 11-02 Research monographs (number theory)

11Axx Elementary number theory

11Mxx Analytic theory of zeta and L-functions

00A07 Problem books

**Keywords: ** problems in number theory; sieves; primes; congruences; barrier for arithmetic function; consecutive primes

**Citations: ** Zbl.451.00001

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